It is known that the directed cycle of order $n$ uniquely achieves the minimum spectral radius among all strongly connected digraphs of order $nge 3$. In this paper, among others, we determine the digraphs which achieve the second, the third and the fourth minimum spectral radii respectively among strongly connected digraphs of order $nge 4$.  

We study the problem of enumerating all rooted directed spanning trees (arborescences) of a directed graph (digraph) G = (V,E) of n vertices. An arborescence A consisting of edges e1, . . . , en−1 can be represented as a monomial e1 · e2 · · · en−1 in variables e ∈ E. All arborescences arb(G) of a digraph then define the Kirchhoff polynomial ∑ A∈arb(G) ∏ e∈A e. We show how to compute a compact ...

Eternal domination is a problem that asks the following question: Can we eternally defend graph ? The principle to against an attacked vertex, changes every turn, by moving guards along edges of graph. In classical version game (Burger et al., 2004), only one guard can move at time, but in m-eternal (Goddard 2005), any number single turn. This led introduction two parameters: eternal and number...

In this paper we investigate reachability relations on the vertices of digraphs. If W is a walk in a digraph D, then the height of W is equal to the number of edges traversed in the direction coinciding with their orientation, minus the number of edges traversed opposite to their orientation. Two vertices u, v ∈ V (D) are Ra,b-related if there exists a walk of height 0 between u and v such that...

A minimum reversing set of a digraph is a smallest sized set of arcs which when reversed makes the digraph acyclic. We investigate a related issue: Given an acyclic digraph D, what is the size of a smallest tournataent T which has the arc set of D as a minimum reversing set? We show that such a T always exists and define the reversing number ofan acyclic digraph to be the number of vertices in ...

We prove that the minimum weight of a dicycle is equal to the maximum number of disjoint dicycle covers, for every weighted digraph whose underlying graph is planar and does not have K5 − e as a minor (K5 − e is the complete graph on five vertices, minus one edge). Equality was previously known when forbidding K4 as a minor, while an infinite number of weighted digraphs show that planarity does...

‎In this paper we defined the vertex removable cycle in respect of the following‎, ‎if $F$ is a class of graphs(digraphs)‎ ‎satisfying certain property‎, ‎$G in F $‎, ‎the cycle $C$ in $G$ is called vertex removable if $G-V(C)in in F $.‎ ‎The vertex removable cycles of eulerian graphs are studied‎. ‎We also characterize the edge removable cycles of regular‎ ‎graphs(digraphs).‎    

Assuming that states will remain a permanent feature of our world, what is the ideal that we should hold out for the international order? An attractive proposal is that those peoples that are already organized under non-dominating, representative states should pursue a twin goal: first, arrange things so that they each enjoy the republican ideal of freedom as non-domination in relation to one a...